A lower bound for $K_{X}L$ of quasi-polarized surfaces $(X,L)$ with non-negative Kodaira dimension

Details A lower bound for $K_{X}L$ of quasi-polarized surfaces $(X,L)$ with non-negative Kodaira dimension A lower bound for $K_{X}L$ of quasi-polarized surfaces $(X,L)$ with non-negative Kodaira dimension

1996-07-15

arxiv.org/abs/alg-geom/9607013v1

Mathematics

Algebraic Geometry

AMS-TeX v2.1, 29pages

Scientific paper

Let $X$ be a smooth projective surface over the complex number field and let $L$ be a nef-big divisor on $X$. Here we consider the following conjecture; If the Kodaira dimension $\kappa(X)\geq 0$, then $K_{X}L\geq 2q(X)-4$, where $q(X)$ is the irregularity of $X$. In this paper, we prove that this conjecture is true if (1) the case in which $\kappa(X)=0$ or 1, (2) the case in which $\kappa(X)=2$ and $h^{0}(L)\geq 2$, or (3) the case in which $\kappa(X)=2$, $X$ is minimal, $h^{0}(L)=1$, and $L$ satisfies some conditions.

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